Representation formulas for solutions of Hamilton-Jacobi equations with convex Hamiltonians. (English) Zbl 1136.35016

Ishii, Hitoshi; Mitake, Hiroyoshi

Indiana Univ. Math. J. 56, No. 5, 2159-2183 (2007).

Summary: We establish general representation formulas for solutions of Hamilton-Jacobi equations with convex Hamiltonians. In order to treat representation formulas on general domains, we introduce notion of ideal boundary similar to the Martin boundary [R. S. Martin, Trans. Am. Math. Soc. 49, 137–172 (1941; Zbl 0025.33302)] in potential theory. We apply such representation formulas to investigate maximal solutions, in certain classes of functions, of Hamilton-Jacobi equations.

MSC:

35F30	Boundary value problems for nonlinear first-order PDEs
35C99	Representations of solutions to partial differential equations
35F20	Nonlinear first-order PDEs

Keywords:

Martin boundary; maximal solutions